Computable topological groups and Pontryagin duality

Alexander Melnikov
2018 Transactions of the American Mathematical Society  
The well-known Pontryagin Duality (classically) reduces the study of compact abelian groups to the algebraic theory of discrete abelian groups. At first glance, Pontryagin Duality seems to be "algorithmic" in nature. Quite unexpectedly, the situation is more intricate. Nonetheless, using methods of computable analysis [Wei00] and modern techniques of computable algebra (e.g., the recent metatheorem [HTMM15]), we establish a partial algorithmic analogy of Pontryagin Dulity and use it to derive a
more » ... handful of applications. We believe that most of these applications are fundamental to the emerging systematic theory of computable Polish groups. We also apply our techniques to measure the complexity of the classification problem for profinite and connected compact Polish groups.
doi:10.1090/tran/7355 fatcat:inuvz2vrvjcphblamqm2fic5m4